Algebraic deformations of toric varieties I. General constructions

Cirio, Lucio S.; Landi, Giovanni; Szabo, Richard J.

doi:10.48550/arxiv.1001.1242

Cited by 6 publications

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“…This paper is the second part of a series of articles devoted to the construction and study of new noncommutative deformations of toric varieties. In the first part [13] the general theory was developed. In the present work we elaborate on and extend some of these developments, and in particular derive a theory of instantons on the noncommutative projective planes CP 2 θ constructed in [13] in several different contexts.…”

Section: Introductionmentioning

confidence: 99%

“…In the first part [13] the general theory was developed. In the present work we elaborate on and extend some of these developments, and in particular derive a theory of instantons on the noncommutative projective planes CP 2 θ constructed in [13] in several different contexts. In the commutative situation, moduli spaces of framed sheaves on the complex projective plane have been studied intensively due to their connection with moduli spaces of framed instantons on the four-sphere; they are the basis for instanton counting.…”

Section: Introductionmentioning

confidence: 99%

“…They may also lead to new examples of BPS states in supersymmetric gauge theories and string theory. See [13] and [37] for further motivation and background behind our constructions.…”

Section: Introductionmentioning

confidence: 99%

“…Outline of paper. In §1 we review and extend the quantization of toric varieties introduced in [13]. Our treatment follows the formalism of cocycle twist quantization and noncommutative geometry in braided monoidal categories, see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In §2 we introduce noncommutative projective varieties following [13], focusing on the noncommutative projective spaces CP n θ . We study coherent sheaves on these varieties and their invariants, and develop a theory of noncommutative monad complexes.…”

Section: Introductionmentioning

confidence: 99%

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Algebraic deformations of toric varieties II: noncommutative instantons

Cirio

Landi

Szabo

2011

Advances in Theoretical and Mathematical Physics

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We continue our study of the noncommutative algebraic and differential geometry of a particular class of deformations of toric varieties, focusing on aspects pertinent to the construction and enumeration of noncommutative instantons on these varieties. We develop a noncommutative version of twistor theory, which introduces a new example of a noncommutative four-sphere. We develop a braided version of the ADHM construction and show that it parameterizes a certain moduli space of framed torsion free sheaves on a noncommutative projective plane. We use these constructions to explicitly build instanton gauge bundles with canonical connections on the noncommutative four-sphere that satisfy appropriate anti-selfduality equations. We construct projective moduli spaces for the torsion free sheaves and demonstrate that they are smooth. We define equivariant partition functions of these moduli spaces, finding that they coincide with the usual instanton partition functions for supersymmetric gauge theories on C 2 .

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…They may also lead to new examples of BPS states in supersymmetric gauge theories and string theory. See [13] and [37] for further motivation and background behind our constructions.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algebraic deformations of toric varieties II: noncommutative instantons

Cirio

Landi

Szabo

2011

Advances in Theoretical and Mathematical Physics

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Mapping spaces and automorphism groups of toric noncommutative spaces

2017

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Curve counting, instantons and McKay correspondences

Cirafici¹,

Szabo

2013

Journal of Geometry and Physics

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a b s t r a c tWe survey some features of equivariant instanton partition functions of topological gauge theories on four and six dimensional toric Kähler varieties, and their geometric and algebraic counterparts in the enumerative problem of counting holomorphic curves. We discuss the relations of instanton counting to representations of affine Lie algebras in the four-dimensional case, and to Donaldson-Thomas theory for ideal sheaves on Calabi-Yau threefolds. For resolutions of toric singularities, an algebraic structure induced by a quiver determines the instanton moduli space through the McKay correspondence and its generalizations. The correspondence elucidates the realization of gauge theory partition functions as quasi-modular forms, and reformulates the computation of noncommutative Donaldson-Thomas invariants in terms of the enumeration of generalized instantons. New results include a general presentation of the partition functions on ALE spaces as affine characters, a rigorous treatment of equivariant partition functions on Hirzebruch surfaces, and a putative connection between the special McKay correspondence and instanton counting on Hirzebruch-Jung spaces.

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Algebraic deformations of toric varieties I. General constructions

Cited by 6 publications

References 0 publications

Algebraic deformations of toric varieties II: noncommutative instantons

Algebraic deformations of toric varieties II: noncommutative instantons

Mapping spaces and automorphism groups of toric noncommutative spaces

Curve counting, instantons and McKay correspondences

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